with \(\widehat{\rho }_g\)  the density matrix of the graph and \(\widehat{\rho }_b(t)\)  the density matrix of the bosons.
This approximation allows us to consider a particle disentangled enough from the graph to be “followed” using the equation (REF ).
It is also a physical requirement,
which accounts for the existence of the particle on its own. In general, we expect that at long times the full hamiltonian thermalizes to a specific graph,
depending on the parameter of the Hamiltonian which defines the metastable state. Later on, we will rescale the coupling constant of the hopping Hamiltonian in order to
obtain the continuum limit. Thus, one could think that this rescaling affects the
state of the graph at infinity. However, the hopping of the bosons allows the graph to thermalize, as it has been shown in [1]}.
Rescaling this constant, just changes the time it takes for the system to thermalize, but not
the asymptotic state of the graph.
As a matter of fact, we do not know yet a Hamiltonian which gives a specific graph state asymptotically. However, the results of [2]} in two dimensions and those
of [3]}, support the conjecture
that, in general, such a Hamiltonian exists. For the time being, it is fair to say that the ground state of Quantum Graphity coupled to a thermal bath are rotational invariant
graphs [4]}. Thus, these graphs can at least be generated by a known effectively 2d-dimensional model.
